Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme

Numerical diffusion and oscillatory behavior characteristics are averted applying numerical solutions of advection-diffusion equation are themselves immensely sophisticated. In this paper, two numerical methods have been used to solve the advection diffusion equation. We use an explicit finite difference scheme for the advection diffusion equation and semi-discretization on the spatial variable for advection-diffusion equation yields a system of ordinary differential equations solved by Euler’s method. Numerical assessment has been executed with specified initial and boundary conditions, for which the exact solution is known. We compare the solutions of the advection diffusion equation as well as error analysis for both schemes.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model.The spatial discretization uses piecewise polynomials of degree k(k≥1)for the stress approximation,degree k+1 for the velocity approximation,and degree k for the numerical trace of velocity on the inter-element boundaries.The temporal discretization in the fully discrete method adopts a backward Euler difference scheme.We show the existence and uniqueness of the semi-discrete and fully discrete solutions,and derive optimal a priori error estimates.Numerical examples are provided to support the theoretical analysis.

SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME

A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving （SSP） Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.

Nonconforming Mixed Finite Element Method for Nonlinear Hyperbolic Equations

A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived for both the displacement and the stress.

AN EFFICIENT THIRD-ORDER SCHEME FOR THREE-DIMENSIONAL HYPERBOLIC CONSERVATION LAWS

In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.

APPROXIMATION OF NONCONFORMING QUASI-WILSON ELEMENT FOR SINE-GORDON EQUATIONS

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

An H^1-Galerkin Nonconforming Mixed Finite Element Method for Integro-Differential Equation of Parabolic Type

H1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type.By use of the typical characteristic of the elements,we obtain that the Galerkin mixed approximations have the same rates of convergence as in the classical mixed method,but without LBB stability condition.

Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices

We consider the drift-diffusion （DD） model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin （LDG） schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit （IMEX） time discretization with the LDG spatial diseretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.

EQ^(rot)_1 Nonconforming Finite Element Method for Nonlinear Dual Phase Lagging Heat Conduction Equations

EQrot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O（h2） one order higher than its interpolation error O（h）, the superclose results of order O（h2） in broken Hi-norm are obtained. At the same time, the global superconvergence in broken Hi-norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O（h4） is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQrot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.

High Accuracy Analysis of the Lowest Order H1-Galerkin Mixed Finite Element Method for Nonlinear Sine-Gordon Equations

The lowest order H1-Galerkin mixed finite element method （for short MFEM） is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart- Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O（h2）/O（h2 ＋ r2） in Hi-norm and H（div; Ω）-norm axe deduced for the semi-discrete and the fully-discrete schemes, where h, r- denote the mesh size and the time step, respectively, which improve the results in the previous literature.

An ultra-accurate dynamic isogeometric analysis with higher order mass formulation

An ultra-accurate isogeometric dynamic analysis is presented.The key ingredient of the proposed methodology is the development of isogeometric higher order mass matrix.A new one-step method is proposed for the construction of higher order mass matrix.In this approach,an adjustable mass matrix is formulated through introducing a set of mass parameters into the consistent mass matrix under the element mass conservation condition.Then the semi-discrete frequency derived from the free vibration equation with the adjustable mass matrix is served as a measure to optimize the mass parameters.In 1D analysis,it turns out that the present one-step method can perfectly recover the existing reduced bandwidth mass matrix and the higher order mass matrix by choosing different mass parameters.However,the employment of the proposed one-step method to the2D membrane problem yields a remarkable gain of solution accuracy compared with the higher order mass matrix generated by the original two-step method.Subsequently a full-discrete isogeometric transient analysis algorithm is presented by using the Newmark time integration scheme and the higher order mass matrix.The full-discrete frequency is derived to assess the accuracy of space-time discretization.Finally a set of numerical examples are presented to evaluate the accuracy of the proposed method,which show that very favorable solution accuracy is achieved by the present dynamic isogeometric analysis with higher order mass formulation compared with that obtained from the standard consistent mass approach.

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward guler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space twogrid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.